Presentation on theme: "From clusters of particles to 2D bubble clusters"— Presentation transcript:
1From clusters of particles to 2D bubble clusters Edwin Flikkema, Simon CoxIMAPS, Aberystwyth University, UK
2Introduction and overview The minimal perimeter problem for 2D equal area bubble clusters.Systems of interacting particlesGlobal optimisation2D particle clusters to 2D bubble clustersVoronoi construction2D particle systems:-log(r) or 1/rp repulsive potentialHarmonic or polygonal confining potentialsResults
32D bubble clusters Minimal perimeter problem: 2D cluster of N bubbles. All bubbles have equal area.Free or confined to the interior of a circle or polygon.Minimize total perimeter (internal + external).Objective: apply techniques used in interacting particle clusters to this minimal perimeter problem.
4Systems of interacting particles System energy:Usually:Example:Lennard-Jones potential:LJ13: Ar13Used in: Molecular Dynamics, Monte Carlo, Energy landscapes
5Energy landscapes Energy vs coordinate Stationary points of U: zero net force on each particleMinima of U correspond to (meta-)stable states.Global minimum is the most stable state.Local optimisation (finding a nearby minimum) relatively easy:Steepest descent, L-BFGS, Powell, etc.Global optimisation: hard.Energy vs coordinateLocal optimisation
6Global optimisation methods Inspired by simulated annealing:Basin hoppingMinima hoppingEvolutionary algorithms:Genetic algorithmOther:Covariance matrix adaptionSimply starting from many random geometries
72D particle systems Energy: Repulsive inter-particle potential: Confining potential:orharmonicpolygonal
82D particle clustersPictures of particle clusters: e.g. N=41, bottom 3 in energy
102D particle clusters Polygonal confining potential: e.g. triangular unit vectorscontour linesdiscontinuous gradient: smoothing needed?
11Technical details List of unique 2D geometries produced Problem: permutational isomers.Distinguishing by energy U not sufficient:Spectrum of inter-particle distances compared.Gradient-based local optimisers have difficulty with polygonal potential due to discontinuous gradientSmoothing needed?Use gradient-less optimisers (e.g. Powell)?
14ConclusionsOptimal geometries of clusters of interacting particles can be used as candidates for the minimal perimeter problem.Various potentials have been tried. 1/r seems to work slightly better than –log(r).Using multiple potentials is recommended.Polygonal potentials have been introduced to represent confinement to a polygon
16Energy landscapes Energy vs coordinate Stationary points of U: zero net force on each particleMinima of U correspond to (meta-)stable states.Global minimum is the most stable state.Saddle points (first order): transition statesNetwork of minima connected by transition statesLocal optimisation (finding a nearby minimum) relatively easy:L-BFGS, Powell, etc.Global optimisation: hard.Local optimisationEnergy vs coordinate
172D clusters: perimeteris fit to data for free clusters